MinT - Best Known (19, 42, s)-Nets in Base 8

Best Known (19, 42, s)-Nets in Base 8

(19, 42, 65)-Net over F₈ — Constructive and digital

Digital (19, 42, 65)-net over F₈, using

t-expansion [i] based on digital (14, 42, 65)-net over F₈, using
- net from sequence [i] based on digital (14, 64)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
    - the Suzuki function field over F₈ [i]

(19, 42, 67)-Net over F₈ — Digital

Digital (19, 42, 67)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(8⁴², 67, F₈, 3, 23) (dual of [(67, 3), 159, 24]-NRT-code), using
- strength reduction [i] based on linear OOA(8⁴², 67, F₈, 3, 24) (dual of [(67, 3), 159, 25]-NRT-code), using
  - constructionÂ X applied to AG(3;F,167P) âŠ‚ AG(3;F,172P) [i] based on
    1. linear OOA(8³⁸, 64, F₈, 3, 24) (dual of [(64, 3), 154, 25]-NRT-code), using algebraic-geometric NRT-code AG(3;F,167P) [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
      - the Suzuki function field over F₈ [i]
    2. linear OOA(8³³, 64, F₈, 3, 19) (dual of [(64, 3), 159, 20]-NRT-code), using algebraic-geometric NRT-code AG(3;F,172P) [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65 (see above)
    3. linear OOA(8⁴, 3, F₈, 3, 4) (dual of [(3, 3), 5, 5]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8⁴, 8, F₈, 3, 4) (dual of [(8, 3), 20, 5]-NRT-code), using
        Reedâ€“Solomon NRT-code RS(3;20,8) [i]

(19, 42, 1622)-Net in Base 8 — Upper bound on s

There is no (19, 42, 1623)-net in base 8, because

1 times m-reduction [i] would yield (19, 41, 1623)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 10 658719 362927 550699 960021 442790 119192 > 8⁴¹ [i]